Tuesday, November 6, 2007 - 1:38 PM

From Batch To Cyclic Process: Optimality And Stability

Yuan Xu, Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 and B. Erik Ydstie, Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213.

Batch and semi-batch processes in a more general framework are examples of cyclic processes. The optimal control of batch processes has been an active research topic for many years. Several methods have been proposed to handle the system nonlinearities and model uncertainty inherited in the batch processes. It is normally difficult to extend current batch and semi-batch control methods to the cyclic processes due to large uncertainty in the initial state. Stability issues have not been fully addressed in the previous literature on the subject.

For the batch process control, several methods have been proposed, including robust optimization (Nagy and Braatz, 2004), MPC-like repeated online estimation-optimization (Ruppen, Bonvin and Rippin, 1997), necessary optimality condition tracking (Srinivasan and Bonvin, 2007) and adaptive extreme-seeking control (Peters, Guay and DeHaan, 2006). Adaptive extreme seeking provides the motivation for our approach. In this work the barrier reformulation to the original problem is used to approach the discontinuous (switching) optimal control profile with a smooth function with error bounds in the L2 norm sense. The continuous profile is parameterized with a novel scheme to handle the large curvatures in the profile. We then analysis the interplay between the parameter adaptation and the gradient based optimum seeking. We further extend the current method to the cyclic processes and derive some preliminary results for the stability analysis of cyclic processes. Two simulation examples have been developed to illustrate the behavior of the proposed control scheme. One example is based on a simple academic test case. The other is based on an experimental system being developed for cyclic production of aluminum using carbothermic reduction. The latter example provides an illustration for how optimal control theory can be used to provide guidelines for optimal process design.

Reference: Nagy Z.K. and R.D. Braatz, (2004), Open-loop and closed-loop robust optimal control of batch processes using distributional and worst-case analysis, Journal of Process Control, 14, 411-422.

Peters N., M. Guay and D. DeHaan, (2006), Real-time dynamic optimization of non-linear batch systems, Canadian Journal of Chemical Engineering, 84, 338-348.

Ruppen D., D. Bonvin and D.W.T. Rippin, (1997), Implementation of adaptive optimal operation for a semi-batch reactive system, Computers and Chemical Engineering, 22, 185-199.

Srinivasan B., D. Bonvin, (2007), Real-Time Optimization of Batch Processes by Tracking the Necessary Conditions of Optimality, Ind. Eng. Chem. Res. 2007, 46, 492-504